Finite Group DatabasebetaFinite Group Databasebeta| order | 1792 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #120 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C7×C8 · Schur multiplier (H2) = C25 |
| direct product | C7×#33432 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C7×C16 | C23 |
| commutator | C22 | C23×C7×C8 |
| Frattini, cosocle | C2×C8 | C24×C7 |
| socle | C22×C7 | C23×C8 |
| prime radical | C7×#525 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C7×C16 ◁ G | |
710 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 448 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3,4,5,6,7)(8,9)(10,11)(12,13,15,20,16,21,25,33,17,19,26,31,27,32,37,41)(14,18,22,29,23,30,34,40,24,28,35,38,36,39,42,43);(9,11);(12,14,15,22,16,23,25,34,17,24,26,35,27,36,37,42)(13,18,20,29,21,30,33,40,19,28,31,38,32,39,41,43);(13,19)(18,28)(20,31)(21,32)(29,38)(30,39)(33,41)(40,43)784 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1792, 26420);